$\sigma_1$ and $\sigma_2$ automorphism of the systems of equations of integrable hierarchies. Discrete and Backlund transformations

TL;DR

This paper explores two automorphisms in integrable hierarchies that generate different system forms, introducing related discrete and Bäcklund transformations and a general multi-soliton solution formula.

Contribution

It identifies two inner automorphisms leading to distinct forms of integrable hierarchy equations and develops associated transformations and multi-soliton solutions.

Findings

Existence of two inner automorphisms for integrable hierarchies

Development of discrete and Bäcklund transformations linked to these automorphisms

Derivation of a general multi-soliton solution formula

Abstract

It is shown that there exists two inner authomorpism which lead to different form of the sistems equations of integrable hierarchy. We present discrete and Backlund transformation connected with such systems and a general formula for multi-soliton solutions based on these symmetries.